The irreligious and Yayin Nesech: Clearing up common misconceptions

Why is wine touched by a non-Jew prohibited? Does this prohibition apply to a non-Jew who’s not an idol worshipper?
Is this extended to a nonreligious Jew who doesn’t keep Shabbos?
What is the status of someone who desecrates Shabbos but goes to shul or makes Kiddush?
What is the law of someone who’s embarrassed to desecrate Shabbos in front of a prominent person?
Do you need witnesses to testify in front of Beis Din to deem someone a Mechalel Shabbos?
Are you allowed to give non-religious Jews wine for them to drink?
Does touching a bottle of wine prohibit the wine?